Package 'MAINT.Data'

Description

MAINT.Data implements methodologies for modelling Interval Data by Normal and Skew-Normal distributions, considering four different possible configurations structures for the variance-covariance matrix. It introduces a data class for representing interval data and includes functions and methods for parametric modelling and analysing of interval data. It performs maximum likelihood and trimmed maximum likelihood estimation, statistical tests, as well as (M)ANOVA, Discriminant Analysis and Gaussian Model Based Clustering.

Details

In the classical model of multivariate data analysis, data is represented in a data-array where n “individuals" (usually in rows) take exactly one value for each variable (usually in columns). Symbolic Data Analysis (see, e.g., Noirhomme-Fraiture and Brito (2011)) provides a framework where new variable types allow to take directly into account variability and/or uncertainty associated to each single “individual", by allowing multiple, possibly weighted, values for each variable. New variable types - interval, categorical multi-valued and modal variables - have been introduced.
We focus on the analysis of interval data, i.e., where elements are described by variables whose values are intervals. Parametric inference methodologies based on probabilistic models for interval variables are developed in Brito and Duarte Silva (2011) where each interval is represented by its midpoint and log-range,for which Normal and Skew-Normal (Azzalini and Dalla Valle (1996)) distributions are assumed. The intrinsic nature of the interval variables leads to special structures of the variance-covariance matrix, which are represented by four different possible configurations.
MAINT.Data implements the proposed methodologies in R, introducing a data class for representing interval data; it includes functions for modelling and analysing interval data, in particular maximum likelihood and trimmed maximum likelihood (Duarte Silva, Filzmoser and Brito (2017)) estimation, and statistical tests for the different considered configurations. Methods for (M)ANOVA, Discriminant Analysis (Duarte Silva and Brito (2015)) and model based clustering (Brito, Duarte Silva and Dias (2015)) of this data class are also provided.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

Maintainer: Pedro Duarte Silva <[email protected]>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P. and Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Brito, P., Duarte Silva, A. P. and Dias, J. G. (2015), Probabilistic Clustering of Interval Data. Intelligent Data Analysis 19(2), 293–313.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Noirhomme-Fraiture, M. and Brito, P. (2011), Far Beyond the Classical Data Models: Symbolic Data Analysis. Statistical Analysis and Data Mining 4(2), 157–170.

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

#Display the first and last observations

head(ChinaT)
tail(ChinaT)

#Print summary statistics

summary(ChinaT)

#Create a new data set considering only the Winter (1st and 4th) quarter intervals

ChinaWT <- ChinaT[,c(1,4)]

# Estimate normal distribution parameters by maximum likelihood, assuming 
# the classical (unrestricted) covariance configuration Case 1

ChinaWTE.C1 <- mle(ChinaWT,CovCase=1)
cat("Winter temperatures of China -- normal maximum likelhiood estimation results:\n")
print(ChinaWTE.C1)
cat("Standard Errors of Estimators:\n") ; print(stdEr(ChinaWTE.C1))

# Estimate normal distribution parameters by maximum likelihood, 
# assuming that one of the C2, C3 or C4 restricted covariance configuration cases hold

ChinaWTE.C234 <- mle(ChinaWT,CovCase=2:4)
cat("Winter temperatures of China -- normal maximum likelihood estimation results:\n")
print(ChinaWTE.C234)
cat("Standard Errors of Estimators:\n") ; print(stdEr(ChinaWTE.C234))

# Estimate normal distribution  parameters robustly by fast maximun trimmed likelihood, 
# assuming that one of the C2, C3 or C4 restricted covariance configuration cases hold

## Not run: 
ChinaWTE.C234 <- fasttle(ChinaWT,CovCase=2:4)
cat("Winter temperatures of China -- normal maximum trimmed likelhiood estimation results:\n")
print(ChinaWTE.C234)

# Estimate skew-normal distribution  parameters 

ChinaWTE.SkN <- mle(ChinaWT,Model="SKNormal")
cat("Winter temperatures of China -- Skew-Normal maximum likelhiood estimation results:\n")
print(ChinaWTE.SkN)
cat("Standard Errors of Estimators:\n") ; print(stdEr(ChinaWTE.SkN))

## End(Not run)

#MANOVA tests assuming that configuration case 1 (unrestricted covariance) 
# or 3 (MidPoints independent of Log-Ranges) holds.  

ManvChinaWT.C13 <- MANOVA(ChinaWT,ChinaTemp$GeoReg,CovCase=c(1,3))
cat("Winter temperatures of China -- MANOVA by geografical regions results:\n")
print(ManvChinaWT.C13)

#Linear Discriminant Analysis

ChinaWT.lda <- lda(ManvChinaWT.C13)
cat("Winter temperatures of China -- linear discriminant analysis results:\n")
print(ChinaWT.lda)
cat("lda Prediction results:\n")
print(predict(ChinaWT.lda,ChinaWT)$class)

## Not run: 
#Estimate error rates by ten-fold cross-validation 

CVlda <- DACrossVal(ChinaWT,ChinaTemp$GeoReg,TrainAlg=lda,
CovCase=BestModel(H1res(ManvChinaWT.C13)),CVrep=1)

#Robust Quadratic Discriminant Analysis

ChinaWT.rqda <- Robqda(ChinaWT,ChinaTemp$GeoReg)
cat("Winter temperatures of China -- robust quadratic discriminant analysis results:\n")
print(ChinaWT.rqda)
cat("robust qda prediction results:\n")
print(predict(ChinaWT.rqda,ChinaWT)$class)

## End(Not run)

# Create an Interval-Data object containing the intervals of loan data
# (from the Kaggle Data Science platform) aggregated by loan purpose

LbyPIdt <- IData(LoansbyPurpose_minmaxDt,
  VarNames=c("ln-inc","ln-revolbal","open-acc","total-acc")) 

print(LbyPIdt)

## Not run: 

#Fit homoscedastic Gaussian mixtures with up to six components

mclustres <- Idtmclust(LbyPIdt,G=1:6)
plotInfCrt(mclustres,legpos="bottomright")
print(mclustres)

#Display the results of the best mixture according to the BIC

summary(mclustres,parameters=TRUE,classification=TRUE)
pcoordplot(mclustres)


## End(Not run)

Description

A interval-valued data set containing 24 units, created from from the Abalone dataset (UCI Machine Learning Repository), after aggregating by sex and age.

Usage

data(Abalone)

Format

AbdaDF: A data frame containing the original 4177 Abalone individuals described by 7 variables.
AbUnits: A factor with 4177 observations and 24 levels indicating the sex by age combination to which each orginal individual belongs to.
AbaloneIdt: An IData object with 24 observations and 7 interval-valued variables, describing the intervals formed by aggregating the AbdaDF microdata by the AbUnits factor.

Description

AgrMcDt creates IData objects by agregating a Data Frame of Micro Data.

Usage

AgrMcDt(MicDtDF, agrby, agrcrt="minmax")

Arguments

Value

An object of class IData with the data set of Interval-valued variables resulting from the aggregation performed.

See Also

IData

Examples

# Create an Interval-Data object by agregating the microdata consisting 
# of 336776 NYC flights included in the FlightsDF data frame, 
# by the statistical units specified in the FlightsUnits factor.

Flightsminmax <- AgrMcDt(FlightsDF,FlightsUnits)

#Display the first and last observations

head(Flightsminmax)
tail(Flightsminmax)

#Print summary statistics

summary(Flightsminmax)

## Not run: 

# Repeat this procedure using now the 10th and 90th percentiles.

Flights1090prcnt <- AgrMcDt(FlightsDF,FlightsUnits,agrcrt=c(0.1,0.9))

#Display the first and last observations

head(Flights1090prcnt)
tail(Flights1090prcnt)

summary(Flights1090prcnt)


## End(Not run)

Description

Selects the best model according to the chosen selection criterion (currently, BIC or AIC)

Usage

BestModel(ModE,SelCrit=c("IdtCrt","BIC","AIC"))

Arguments

Value

An integer with the index of the model chosen by the selection criterion

Description

This data set consist of the intervals for four characteristics (Price, EngineCapacity, TopSpeed and Acceleration) of 27 cars models partitioned into four different classes (Utilitarian, Berlina, Sportive and Luxury).

Usage

data(Cars)

Format

A data frame containing 27 observations on 9 variables, the first eight with the the lower and upper bounds of the interval characteristics for 27 car models, the last one a factor indicating the model class.

Description

This data set consist of the intervals of observed temperatures (Celsius scale) in each of the four quarters, Q_1 to Q_4, of the years 1974 to 1988 in 60 chinese meteorologic stations; one outlier observation (YinChuan_1982) has been discarded. The 60 stations belong to different regions in China, which therefore define a partition of the 899 stations-year combinations.

Usage

data(ChinaTemp)

Format

A data frame containing 899 observations on 9 variables, the first eight with the lower and upper bounds of the temperatures by quarter in the 899 stations-year combinations, the last one a factor indicating the geographic region of each station.

Description

S4 methods for function coef. As in the generic coef S3 ‘stats’ method, these methods extract parameter estimates for the models fitted to Interval Data.

Usage

## S4 method for signature 'IdtNDE'
coef(object, selmodel=BestModel(object), ...)
## S4 method for signature 'IdtSNDE'
coef(object, selmodel=BestModel(object), ParType=c("Centr", "Direct", "All"), ...)
## S4 method for signature 'IdtNandSNDE'
coef(object, selmodel=BestModel(object),  ParType=c("Centr", "Direct", "All"), ...)

Arguments

Value

A list of parameter estimates. The list components depend on the model and parametriztion assumed by the model. For Gaussian models these are respectivelly mu (vector of mean estimates) and Sigma (matrix of covariance estimates). For Skew-Normal models the components are mu, Sigma and gamma1 (one vector of skewness coefficient estimates) for the centred parametrization and the vectors ksi, and alpha, and the matrix Omega for the direct parametrization.

References

Arellano-Valle, R. B. and Azzalini, A. (2008): "The centred parametrization for the multivariate skew-normal distribution". Journal of Multivariate Analysis, Volume 99, Issue 7, 1362-1382.

See Also

stdEr, vcov

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

ChinaT_NE <- mle(ChinaT)

# Display model estimates

print(coef(ChinaT_NE))

## Not run: 

# Estimate Skew-Normal distribution  parameters by maximum likelihood  

ChinaT_SNE <- mle(ChinaT,Model="SKNormal")

# Display model estimates

print(coef(ChinaT_SNE,ParType="Centr"))
print(coef(ChinaT_SNE,ParType="Direct"))


## End(Not run)

Description

‘ConfMat’ creates confussion matrices from two factor describing, respectively, original classes and predicted classification results

Usage

ConfMat(origcl, predcl, otp=c("absandrel","abs","rel"), dec=3)

Arguments

Value

When argument ‘otp’ is set to “absandrel” (default), a list with two confusion matrices, respectively with absolute and relative frequencies. When argument ‘otp’ is set to “abs” a confusion matrix with absolute frequencies, and when argument ‘otp’ is set to “rel” a confusion matrix with relative frequencies.

Author(s)

A. Pedro Duarte Silva

See Also

lda, qda, snda, Roblda, Robqda, DACrossVal

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

#Linear Discriminant Analysis

ChinaT.lda <- lda(ChinaT,ChinaTemp$GeoReg)
ldapred <- predict(ChinaT.lda,ChinaT)$class

# lda resubstitution confusion matrix

ConfMat(ChinaTemp$GeoReg,ldapred)

#Quadratic Discriminant Analysis

ChinaT.qda <- qda(ChinaT,ChinaTemp$GeoReg)
qdapred <- predict(ChinaT.qda,ChinaT)$class

# qda resubstitution confusion matrix

ConfMat(ChinaTemp$GeoReg,qdapred)

Description

ConfTests contains a list of the results of statistical likelihood-ratio tests that evaluate the goodness-of-fit of restricted models against more general ones. Currently, the models implemented are those based on the Normal and Skew-Normal distributions, with the four alternative variance-covariance matrix configurations.

Slots

TestRes:

List of test results; each element is an object of class LRTest, with the following components:

ChiSq: Value of the Chi-Square statistics corresponding to the performed test.
df: Degrees of freedom of the Chi-Square statistics.
pvalue: p-value of the Chi-Square statistics value, obtained from the Chi-Square distribution with df degrees of freedom.
H0logLik: Logarithm of the Likelihood function under the null hypothesis.
H1logLik: Logarithm of the Likelihood function under the alternative hypothesis.

RestModels:

The restricted model (corresponding to the null hypothesis)

FullModels:

The full model (corresponding to the alternative hypothesis)

Methods

show

signature(object = "ConfTests"): show S4 method for the ConfTests-class

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

See Also

mle, IData, LRTest

Description

S4 methods for function cor. These methods extract estimates of correlation matrices for the models fitted to Interval Data.

Usage

## S4 method for signature 'IdtNDE'
cor(x)
## S4 method for signature 'IdtSNDE'
cor(x)
## S4 method for signature 'IdtNandSNDE'
cor(x)
## S4 method for signature 'IdtMxNDE'
cor(x)
## S4 method for signature 'IdtMxSNDE'
cor(x)

Arguments

Value

For the IdtNDE, IdtSNDE and IdtNandSNDE methods or IdtMxNDE, IdtMxSNDE methods with slot “Hmcdt” equal to TRUE: a matrix with the estimated correlations.
For the IdtMxNDE, and IdtMxSNDE methods with slot “Hmcdt” equal to FALSE: a three-dimensional array with a matrix with the estimated correlations for each group at each level of the third dimension.

See Also

var

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

ChinaT_NE <- mle(ChinaT)

# Display correlation estimates

print(cor(ChinaT_NE))

Description

‘DACrossVal’ evaluates the performance of a Discriminant Analysis training sample algorithm by k-fold Cross-Validation.

Usage

DACrossVal(data, grouping, TrainAlg, EvalAlg=EvalClrule, 
Strfolds=TRUE, kfold=10, CVrep=20, prior="proportions", loo=FALSE, dec=3, ...)

Arguments

Value

A three dimensional array with the number of tested observations, and estimated classification errors for each combination of fold and replication tried. The array dimensions are defined as follows:
The first dimension runs through the different fold-replication combinations.
The second dimension represents the classes.
The third dimension has two named levels representing respectively the number of observations tested (“Nk”), and the estimated classification errors (“Clerr”).

Author(s)

A. Pedro Duarte Silva

See Also

lda, qda, IData

Examples

## Not run: 

# Compare performance of linear and quadratic discriminant analysis with 
#  Covariance cases C1 and c4 on the ChinaT data set by 5-fold cross-validation 
#  replicated twice

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8])

# Classical (configuration 1) Linear Discriminant Analysis 

CVldaC1 <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=lda,CovCase=1,kfold=5,CVrep=2)
summary(CVldaC1[,,"Clerr"])

# Linear Discriminant Analysis with covariance case 3

CVldaC4 <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=lda,CovCase=3,kfold=5,CVrep=2)
summary(CVldaC4[,,"Clerr"])

# Classical (configuration 1) Quadratic Discriminant Analysis 

CVqdaC1 <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=qda,CovCase=1,kfold=5,CVrep=2)
summary(CVqdaC1[,,"Clerr"])

# Quadratic Discriminant Analysis with covariance case 3

CVqdaC4 <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=qda,CovCase=3,kfold=5,CVrep=2)
summary(CVqdaC4[,,"Clerr"])


## End(Not run)

Description

This function will create a control object of class EMControl containing the control parameters for the EM algorithm used in estimation of Gaussian mixtures by function Idtmclust.

Usage

EMControl(nrep=0, maxiter=1000, convtol=0.01, protol=1e-3, seed=NULL, pertubfct=1, 
   k2max=1e6, MaxVarGRt=1e6)

Arguments

Value

An EMControl object

See Also

Idtmclust

Description

This class contains the control parameters for the EM algorithm used in estimation of Gaussian mixtures by function Idtmclust. .

Objects from the Class

Objects can be created by calls of the form new("EMControl", ...) or by calling the constructor-function EMControl.

Slots

nrep

Number of replications (different randomly generated starting points) of the EM algorithm.

maxiter

Maximum number of iterations in each replication of the EM algorithm.

convtol

Numeric tolerance for testing the convergence of the EM algorithm. Convergence is assumed when the log-likelihood changes less than convtol.

protol

Numeric tolerance for the mixture proportions. Proportions below protol, considered to be zero, are not allowed.

seed

Starting value for random generator.

See Also

EMControl

Description

“extmatrix” is a simple extension of the base matrix class, that that accepts NULL objects as members.

Extends

Class matrix, directly.

Description

Performs maximum trimmed likelihood estimation by the fasttle algorithm

Usage

fasttle(Sdt,
    CovCase=1:4,
    SelCrit=c("BIC","AIC"),
    alpha=control@alpha,
    nsamp = control@nsamp,
    seed=control@seed,
    trace=control@trace,
    use.correction=control@use.correction,
    ncsteps=control@ncsteps,
    getalpha=control@getalpha,
    rawMD2Dist=control@rawMD2Dist,				
    MD2Dist=control@MD2Dist,
    eta=control@eta,
    multiCmpCor=control@multiCmpCor,				
    getkdblstar=control@getkdblstar,
    outlin=control@outlin,
    trialmethod=control@trialmethod,
    m=control@m,
    reweighted = control@reweighted,
    k2max = control@k2max, 
    otpType=control@otpType,
    control=RobEstControl(), ...)

Arguments

Value

An object of class IdtE with the fasttle estimates, the value of the comparison criterion used to select the covariance configurations, the robust squared Mahalanobis distances, and optionally (if argument ‘otpType’ is set to true) performance statistics concerning the algorithm execution.

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Hadi, A. S. and Luceno, A. (1997), Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Computational Statistics and Data Analysis 25(3), 251–272.

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

Todorov V. and Filzmoser P. (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software 32(3), 1–47.

See Also

fulltle, RobEstControl, getIdtOutl, IdtSngNDRE

Examples

## Not run: 

# Create an Interval-Data object containing the intervals of temperatures by quarter 
# for 899 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8])

# Estimate parameters by the fast trimmed maximum likelihood estimator, 
# using a two-step procedure to select the trimming parameter, a reweighted 
# MCD estimate, and the classical 97.5% chi-square quantile cut-offs.

Chinafasttle1 <- fasttle(ChinaT)
cat("China maximum trimmed likelihood estimation results =\n")
print(Chinafasttle1)

# Estimate parameters by the fast trimmed maximum likelihood estimator, using 
# the triming parameter that maximizes breakdown, and a reweighted MCD estimate 
# based on the 97.5% quantiles of Hardin and Rocke adjusted F distributions.

Chinafasttle2 <- fasttle(ChinaT,alpha=0.5,getalpha=FALSE,rawMD2Dist="HardRockeAdjF")
cat("China maximum trimmed likelihood estimation results =\n")
print(Chinafasttle2)

# Estimate parameters by the fast trimmed maximum likelihood estimator, using a two-step procedure
# to select the triming parameter, a reweighed MCD estimate based on Hardin and Rocke adjusted 
# F distributions, and 95% quantiles, and the Cerioli Beta and F distributions together
# with Cerioli iterated procedure to identify outliers in the first step.

Chinafasttle3 <- fasttle(ChinaT,rawMD2Dist="HardRockeAdjF",eta=0.05,MD2Dist="CerioliBetaF",
multiCmpCor="iterstep")
cat("China maximum trimmed likelihood estimation results =\n")
print(Chinafasttle3)


## End(Not run)

Description

Performs maximum trimmed likelihood estimation by an exact algorithm (full enumeratiom of all k-trimmed subsets)

Usage

fulltle(Sdt, CovCase = 1:4, SelCrit = c("BIC", "AIC"), alpha =
                 0.75, use.correction = TRUE, getalpha = "TwoStep",
                 rawMD2Dist = c("ChiSq", "HardRockeAsF",
                 "HardRockeAdjF"), MD2Dist = c("ChiSq",
                 "CerioliBetaF"), eta = 0.025, multiCmpCor = c("never",
                 "always", "iterstep"), outlin = c("MidPandLogR",
                 "MidP", "LogR"), reweighted = TRUE, k2max=1e6, 
                 force = FALSE, ...)

Arguments

Value

An object of class IdtE with the fulltle estimates, the value of the comparison criterion used to select the covariance configurations and the robust squared Mahalanobis distances.

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Hadi, A. S. and Luceno, A. (1997), Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Computational Statistics and Data Analysis 25(3), 251–272.

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

See Also

fasttle, getIdtOutl

Examples

## Not run: 

# Create an Interval-Data object containing the intervals for characteristics 
# of 27 cars models.

CarsIdt <- IData(Cars[1:8],VarNames=c("Price","EngineCapacity","TopSpeed","Acceleration"))

# Estimate parameters by the full trimmed maximum likelihood estimator, 
# using a two-step procedure to select the trimming parameter, a reweighed 
# MCD estimate, and the classical 97.5% chi-square quantile cut-offs.

CarsTE1 <- fulltle(CarsIdt)
cat("Cars data -- normal maximum trimmed likelihood estimation results:\n")
print(CarsTE1)
		
# Estimate parameters by the full trimmed maximum likelihood estimator, using
# the triming parameter that maximizes breakdown, and a reweighed MCD estimate
# based on the 97.5% quantiles of Hardin and Rocke adjusted F distributions.
		
CarsTE2 <- fulltle(CarsIdt,alpha=0.5,getalpha=FALSE,rawMD2Dist="HardRockeAdjF")
cat("Cars data -- normal maximum trimmed likelihood estimation results:\n")
print(CarsTE2)
		
# Estimate parameters by the full trimmed maximum likelihood estimator, using 
# a two-step procedure to select the trimming parameter, and a reweighed MCD estimate 
# based on Hardin and Rocke adjusted F distributions, 95% quantiles, and 
# the Cerioli Beta and F distributions together with his iterated procedure 
# to identify outliers in the first step.
		
CarsTE3 <- fulltle(CarsIdt,rawMD2Dist="HardRockeAdjF",eta=0.05,MD2Dist="CerioliBetaF",
multiCmpCor="iterstep")
cat("Cars data -- normal maximum trimmed likelihood estimation results:\n")
print(CarsTE3)


## End(Not run)

Description

Identifies outliers in a data set of Interval-valued variables

Usage

getIdtOutl(Sdt, IdtE=NULL, muE=NULL, SigE=NULL,
  eta=0.025, Rewind=NULL, m=length(Rewind),
  RefDist=c("ChiSq","HardRockeAdjF","HardRockeAsF","CerioliBetaF"),
  multiCmpCor=c("never","always","iterstep"), 
  outlin=c("MidPandLogR","MidP","LogR"))

Arguments

Value

A vector with the indices of the entities identified as outliers.

References

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

See Also

fasttle, fulltle

Examples

## Not run: 

# Create an Interval-Data object containing the intervals for characteristics 
# of 27 cars models.

CarsIdt <- IData(Cars[1:8],VarNames=c("Price","EngineCapacity","TopSpeed","Acceleration"))

# Estimate parameters by the fast trimmed maximum likelihood estimator, 
# using a two-step procedure to select the trimming parameter, a reweighed 
# MCD estimate, and the classical 97.5% chi-squared quantile cut-offs.
			
Carstle1 <- fulltle(CarsIdt)
			
# Get and display the outliers using the classical 97.5% chi-squared quantile cut-offs.
		
CarsOtl1 <- getIdtOutl(CarsIdt,Carstle1)
print(CarsOtl1)
plot(CarsOtl1)
			
# Estimate parameters by the fast trimmed maximum likelihood estimator, 
# using a two-step procedure to select the trimming parameter, and a reweighed  
# based on the 97.5% quantiles of Hardin and Rocke adjusted F distributions.
			
Carstle2 <- fulltle(CarsIdt,rawMD2Dist="HardRockeAdjF")
			
# Get and display the outliers using the 97.5
			
CarsTtl2 <- getIdtOutl(CarsIdt,Carstle2,RefDist="CerioliBetaF")
print(CarsTtl2)
plot(CarsTtl2)



## End(Not run)

Description

IData creates IData objects from data frames of interval bounds or MidPoint/LogRange values of the interval-valued observations.

Usage

IData(Data, 
Seq = c("LbUb_VarbyVar", "MidPLogR_VarbyVar", "AllLb_AllUb", "AllMidP_AllLogR"), 
VarNames=NULL, ObsNames=row.names(Data), NbMicroUnits=integer(0))

Arguments

Details

Objects of class IData describe a data set of ‘NObs’ observations on ‘NIVar’ Interval-valued variables. This function creates an interval-data object from a data-frame with either the lower and upper bounds of the observed intervals or by their midpoints and log-ranges.

See Also

IData, AgrMcDt

Examples

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))
cat("Summary of the ChinaT IData object:\n")  ; print(summary(ChinaT))
cat("ChinaT first ant last three observations:\n")  
print(head(ChinaT,n=3))
cat("\n...\n")
print(tail(ChinaT,n=3))

Description

A data-array of interval-valued data is an array where each of the NObs rows, corresponding to each entity under analysis, contains the observed intervals of the NIVar descriptive variables.

Slots

MidP:

A data-frame of the midpoints of the observed intervals

LogR:

A data-frame of the logarithms of the ranges of the observed intervals

ObsNames:

An optional vector of names assigned to the individual observations.

VarNames:

An optional vector of names to be assigned to the Interval-valued Variables.

NObs:

Number of entities under analysis (cases)

NIVar:

Number of interval variables

NbMicroUnits:

An integer vector with the number of micro data units by interval-valued observation (or an empty vector, if not applicable)

Methods

show

signature(object = "IData"): show S4 method for the IData-class.

nrow

signature(x = "IData"): returns the number of statistical units (observations).

ncol

signature(x = "IData"): returns the number of of Interval-valued variables.

dim

signature(x = "IData"): returns a vector with the of number statistical units as first element, and the number of Interval-valued variables as second element.

rownames

signature(x = "IData"): returns the row (entity) names for an object of class IData.

colnames

signature(x = "IData"): returns column (variable) names for an object of class IData.

names

signature(x = "IData"): returns column (variable) names for an object of class IData.

MidPoints

signature(Sdt = "IData"): returns a data frame with MidPoints for an object of class IData.

LogRanges

signature(Sdt = "IData"): returns a data frame with LogRanges for an object of class IData.

Ranges

signature(Sdt = "IData"): returns an data frame with Ranges for an object of class IData.

NbMicroUnits

signature(Sdt = "IData"): returns an integer vector with the number of micro data units by interval-valued observation for an object of class IData.

head

signature(x = "IData"): head S4 method for the IData-class.

tail

signature(x = "IData"): tail S4 method for the IData-class.

plot

signature(x = "IData"): plot S4 methods for the IData-class.

mle

signature(x = "IData"): Maximum likelihood estimation.

fasttle

signature(x = "IData"): Fast trimmed maximum likelihood estimation.

fulltle

signature(x = "IData"): Exact trimmed maximum likelihood estimation.

RobMxtDEst

signature(x = "IData"): Robust estimation of distribution mixtures for interval-valued data.

MANOVA

signature(x = "IData"): MANOVA tests on the interval-valued data.

lda

signature(x = "IData"): Linear Discriminant Analysis using maximum likelihood parameter estimates of Gaussian mixtures.

qda

signature(x = "IData"): Quadratic Discriminant Analysis using maximum likelihood parameter estimates of Gaussian mixtures.

Roblda

signature(x = "IData"): Linear Discriminant Analysis using robust estimates of location and scatter.

Robqda

signature(x = "IData"): Quadratic Discriminant Analysis using robust estimates of location and scatter.

snda

signature(x = "IData"): Discriminant Analysis using maximum likelihood parameter estimates of SkewNormal mixtures.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Noirhomme-Fraiture, M., Brito, P. (2011), Far Beyond the Classical Data Models: Symbolic Data Analysis. Statistical Analysis and Data Mining 4(2), 157–170.

See Also

IData, AgrMcDt, mle, fasttle, fulltle, RobMxtDEst, MANOVA, lda, qda, Roblda, Robqda

Description

IdtE contains estimation results for the models assumed for single distributions, or mixtures of distributions, underlying data sets of interval-valued entities.

Slots

ModelNames:

The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration (Case 1 through Case 4)

ModelType:

Indicates the model; currently, Gaussian or Skew-Normal distributions are implemented

ModelConfig:

Configuration of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Number of interval variables

SelCrit:

The model selection criterion; currently, AIC and BIC are implemented

logLiks:

The logarithms of the likelihood function for the different cases

AICs:

Value of the AIC criterion

BICs:

Value of the BIC criterion

BestModel:

Bestmodel indicates the best model according to the chosen selection criterion

SngD:

Boolean flag indicating whether a single or a mixture of distribution were estimated

Methods

BestModel

signature(Sdt = "IdtE"): Selects the best model according to the chosen selection criterion (currently, AIC or BIC)

show

signature(object = "IdtE"): show S4 method for the IDtE-class

summary

signature(object = "IdtE"): summary S4 method for the IDtE-class

testMod

signature(Sdt = "IdtE"): Performs statistical likelihood-ratio tests that evaluate the goodness-of-fit of a nested model against a more general one.

sd

signature(Sdt = "IdtE"): extracts the standard deviation estimates from objects of class IdtE.

AIC

signature(Sdt = "IdtE"): extracts the value of the Akaike Information Criterion from objects of class IdtE.

BIC

signature(Sdt = "IdtE"): extracts the value of the Bayesian Information Criterion from objects of class IdtE.

logLik

signature(Sdt = "IdtE"): extracts the value of the maximised log-likelihood from objects of class IdtE.

mean

signature(x = "IdtE"): extracts the mean vector estimate from objects of class IdtE

var

signature(x = "IdtE"): extracts the variance-covariance matrix estimate from objects of class IdtE

cor

signature(x = "IdtE"): extracts the correlation matrix estimate from objects of class IdtE

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

mle, fasttle, fulltle, MANOVA, RobMxtDEst, IData

Description

Idtlda contains the results of Linear Discriminant Analysis for the interval data

Slots

prior:

Prior probabilities of class membership; if unspecified, the class proportions for the training set are used; if present, the probabilities should be specified in the order of the factor levels.

means:

Matrix with the mean vectors for each group

scaling:

Matrix which transforms observations to discriminant functions, normalized so that within groups covariance matrix is spherical.

N:

Number of observations

CovCase:

Configuration case of the variance-covariance matrix: Case 1 through Case 4

Methods

predict

signature(object = "Idtlda"): Classifies interval-valued observations in conjunction with lda.

show

signature(object = "Idtlda"): show S4 method for the IDdtlda-class

CovCase

signature(object = "Idtlda"): Returns the configuration case of the variance-covariance matrix

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

qda, MANOVA, Roblda, Robqda, snda, IData

Description

IdtMANOVA extends LRTest directly, containing the results of MANOVA tests on the interval-valued data. This class is not used directly, but is the basis for different specializations according to the model assumed for the distribution in each group. In particular, the following specializations of IdtMANOVA are currently implemented:

IdtClMANOVA extends IdtMANOVA, assuming a classical (i.e., homoscedastic gaussian) setup.

IdtHetNMANOVA extends IdtMANOVA, assuming a heteroscedastic gaussian set-up.

IdtLocSNMANOVA extends IdtMANOVA, assuming a Skew-Normal location model set-up.

IdtLocNSNMANOVA extends IdtMANOVA, assuming either a homoscedastic gaussian or Skew-Normal location model set-up.

IdtGenSNMANOVA extends IdtMANOVA, assuming a Skew-Normal general model set-up.

IdtGenNSNMANOVA extends IdtMANOVA, assuming either a heteroscedastic gaussian or Skew-Normal general model set-up.

Slots

NIVar:

Number of interval variables.

grouping:

Factor indicating the group to which each observation belongs to.

H0res:

Model estimates under the null hypothesis.

H1res:

Model estimates under the alternative hypothesis.

ChiSq:

Inherited from class LRTest. Value of the Chi-Square statistics corresponding to the performed test.

df:

Inherited from class LRTest. Degrees of freedom of the Chi-Square statistics.

pvalue:

Inherited from class LRTest. p-value of the Chi-Square statistics value, obtained from the Chi-Square distribution with df degrees of freedom.

H0logLik:

Inherited from class LRTest. Logarithm of the Likelihood function under the null hypothesis.

H1logLik:

Inherited from class LRTest. Logarithm of the Likelihood function under the alternative hypothesis.

Methods

show

signature(object = "IdtMANOVA"): show S4 method for the IdtMANOVA-classes.

summary

signature(object = "IdtMANOVA"): summary S4 method for the IdtMANOVA-classes.

H0res

signature(object = "IdtMANOVA"): retrieves the model estimates under the null hypothesis.

H1res

signature(object = "IdtMANOVA"): retrieves the model estimates under the alternative hypothesis.

lda

signature(x = "IdtClMANOVA"): Linear Discriminant Analysis using the estimated model parameters.

lda

signature(x = "IdtLocNSNMANOVA"): Linear Discriminant Analysis using the estimated model parameters.

qda

signature(x = "IdtHetNMANOVA"): Quadratic Discriminant Analysis using the estimated model parameters.

qda

signature(x = "IdtGenNSNMANOVA"): Quadratic Discriminant Analysis using the estimated model parameters.

snda

signature(x = "IdtLocNSNMANOVA"): Discriminant Analysis using maximum likelihood parameter estimates of SkewNormal mixtures assuming a "location" model (i.e., groups differ only in location parameters).

snda

signature(x = "IdtGenSNMANOVA"): Discriminant Analysis using maximum likelihood parameter estimates of SkewNormal mixtures assuming a general model (i.e., groups differ in all parameters).

snda

signature(x = "IdtGenNSNMANOVA"): Discriminant Analysis using maximum likelihood parameter estimates of SkewNormal mixtures assuming a general model (i.e., groups differ in all parameters).

Extends

Class LRTest, directly.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012): "Modelling Interval Data with Normal and Skew-Normal Distributions". Journal of Applied Statistics, Volume 39, Issue 1, 3-20.

See Also

MANOVA, lda, qda, snda, IData

Description

IdtMclust contains the results of fitting mixtures of Gaussian distributions to interval data represented by objects of class IData.

Slots

call:

The matched call that created the IdtMclust object

data:

The IData data object

NObs:

Number of entities under analysis (cases)

NIVar:

Number of interval variables

SelCrit:

The model selection criterion; currently, AIC and BIC are implemented

Hmcdt:

Indicates whether the optimal model corresponds to a homoscedastic (TRUE) or a hetereocedasic (FALSE) setup

BestG:

The optimal number of mixture components.

BestC:

The configuration case of the variance-covariance matrix in the optimal model

logLiks:

The logarithms of the likelihood function for the different models tried

logLik:

The logarithm of the likelihood function for the optimal model

AICs:

The values of the AIC criterion for the different models tried

aic:

The value of the AIC criterion for the he optimal model

BICs:

The values of the BIC criterion for the different models tried

bic:

The value of the BIC criterion for the he optimal model

parameters

A list with the following components:

pro

A vector whose kth component is the mixing proportion for the kth component of the mixture model.

mean

The mean for each component. If there is more than one component, this is a matrix whose kth column is the mean of the kth component of the mixture model.

covariance

A three-dimensional array with the covariance estimates. If Hmcdt is FALSE (heteroscedastic setups) the third dimension levels run through the BestG mixture components, with one different covariance matrix for each level. Otherwise (homoscedastic setups), there is only one covariance matrix and the size of the third dimension equals one.

z:

A matrix whose [i,k]th entry is the probability that observation i in the test data belongs to the kth class.

classification:

The classification corresponding to z, i.e. map(z).

allres:

A list with the detailed results for all models fitted.

Methods

show

signature(object = "IdtMclust"): show S4 method for the IdtMclust-class

summary

signature(object = "IdtMclust"): summary S4 method for the IdtMclust-class

parameters

signature(x = "IdtMclust"): retrieves the value of the parameter estimates for the obtained partition

pro

signature(x = "IdtMclust"): retrieves the value of the estimated mixing proportions for the obtained partition

mean

signature(x = "IdtMclust"): retrieves the value of the component means for the obtained partition

var

signature(x = "IdtMclust"): retrieves the value of the estimated covariance matrices for the obtained partition

cor

signature(x = "IdtMclust"): retrieves the value of the estimated correlation matrices

classification

signature(x = "IdtMclust"): retrieves the individual class assignments for the obtained partition

SelCrit

signature(x = "IdtMclust"): retrieves a string specifying the criterion used to find the best model and partition

Hmcdt

signature(x = "IdtMclust"): returns TRUE if an homecedastic model has been assumed, and FALSE otherwise

BestG

signature(x = "IdtMclust"): returns the number of components selectd

BestC

signature(x = "IdtMclust"): retruns the covariance configuration selected

PostProb

signature(x = "IdtMclust"): retrieves the estimates of the individual posterir probabilities for the obtained partition

BIC

signature(x = "IdtMclust"): returns the value of the BIC criterion

AIC

signature(x = "IdtMclust"): returns the value of the AIC criterion

logLik

signature(x = "IdtMclust"): returns the value of the log-likelihood

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Brito, P., Duarte Silva, A. P. and Dias, J. G. (2015), Probabilistic Clustering of Interval Data. Intelligent Data Analysis 19(2), 293–313.

See Also

Idtmclust, plotInfCrt, pcoordplot

Description

Performs Gaussian model based clustering for interval data

Usage

Idtmclust(Sdt, G = 1:9, CovCase=1:4, SelCrit=c("BIC","AIC"),
  Mxt=c("Hom","Het","HomandHet"), control=EMControl())

Arguments

Value

An object of class IdtMclust providing the optimal (according to BIC) mixture model estimation.

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Brito, P., Duarte Silva, A. P. and Dias, J. G. (2015), Probabilistic Clustering of Interval Data. Intelligent Data Analysis 19(2), 293–313.

Fraley, C., Raftery, A. E., Murphy, T. B. and Scrucca, L. (2012), mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington.

See Also

IdtMclust, EMControl, EMControl, plotInfCrt, pcoordplot

Examples

## Not run: 

# Create an Interval-Data object containing the intervals of loan data
# (from the Kaggle Data Science platform) aggregated by loan purpose

LbyPIdt <- IData(LoansbyPurpose_minmaxDt,
                 VarNames=c("ln-inc","ln-revolbal","open-acc","total-acc")) 

print(LbyPIdt)

#Fit homoscedastic Gaussian mixtures with up to nine components

mclustres <- Idtmclust(LbyPIdt)
plotInfCrt(mclustres,legpos="bottomright")
print(mclustres)

#Display the results of the best mixture according to the BIC

summary(mclustres,parameters=TRUE,classification=TRUE)
pcoordplot(mclustres)

#Repeat the analysus with both homoscedastic and heteroscedastic mixtures up to six components

mclustres1 <- Idtmclust(LbyPIdt,G=1:6,Mxt="HomandHet")
plotInfCrt(mclustres1,legpos="bottomright")
print(mclustres1)

#Display the results of the best heteroscedastic mixture according to the BIC

summary(mclustres1,parameters=TRUE,classification=TRUE,model="HetG2C2")


## End(Not run)

Description

IdtMxE extends the IdtE class, assuming that the data can be characterized by a mixture of distributions, for instances considering partitions of entities into different groups.

Slots

grouping:

Factor indicating the group to which each observation belongs to

ModelNames:

Inherited from class IdtE. The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; currently, Gaussian or Skew-Normal distributions are implemented.

ModelConfig:

Inherited from class IdtE. Configuration of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Bestmodel indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to FALSE in objects of class "IdtMxE"

Ngrps:

Number of mixture components

Extends

Class IdtE, directly.

Methods

No methods defined with class "IdtMxE" in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IdtE, IdtSngDE, IData, MANOVA, RobMxtDEst

Description

IdtMxNandSNDE contains the results of a mixture model estimation; Normal an Skew-Normal models are considered, with the four different possible variance-covariance configurations.

Slots

NMod:

Estimates of the mixture model for the Gaussian case

SNMod:

Estimates of the mixture model for the Skew-Normal case

grouping:

Inherited from class IdtMxE. Factor indicating the group to which each observation belongs to

ModelNames:

Inherited from class IdtE. The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; currently, Gaussian or Skew-Normal distributions are implemented

ModelConfig:

Inherited from class IdtE. Configuration case of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to FALSE in objects of class IdtMxNandSNDE

Ngrps:

Inherited from class IdtMxE. Number of mixture components

Extends

Class IdtMxE, directly. Class IdtE, by class IdtMxE, distance 2.

Methods

No methods defined with class IdtMxNandSNDE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IdtE, IdtMxE, IdtSngNandSNDE, MANOVA, RobMxtDEst, IData

Description

IdtMxNDE contains the results of a mixture Normal model maximum likelihood parameter estimation, with the four different possible variance-covariance configurations.

Slots

Hmcdt:

Indicates whether we consider an homocedastic (TRUE) or a hetereocedasic model (FALSE)

mleNmuE:

Matrix with the maximum likelihood mean vectors estimates by group (each row refers to a group)

mleNmuEse:

Matrix with the maximum likelihood means' standard errors by group (each row refers to a group)

CovConfCases:

List of the considered configurations

grouping:

Inherited from class IdtMxE. Factor indicating the group to which each observation belongs to

ModelNames:

Inherited from class IdtE. The model acronym formed by a "N", indicating a Normal model, followed by the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; always set to "Normal" in objects of the IdtMxNDE class

ModelConfig:

Inherited from class IdtE. Configuration case of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to FALSE in objects of class IdtMxNDE

Ngrps:

Inherited from class IdtMxE. Number of mixture components

Extends

Class IdtMxE, directly. Class IdtE, by class IdtMxE, distance 2.

Methods

lda

signature(x = "IdtMxtNDE"): Linear Discriminant Analysis using the estimated model parameters.

qda

signature(x = "IdtMxtNDE"): Quadratic Discriminant Analysis using the estimated model parameters.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IdtE, IdtMxE, IdtMxNDRE, IdtSngNDE, IData, MANOVA

Description

IdtMxNDRE contains the results of a mixture Normal model robust parameter estimation, with the four different possible variance-covariance configurations.

Slots

Hmcdt:

Indicates whether we consider an homocedastic (TRUE) or a hetereocedasic model (FALSE)

RobNmuE:

Matrix with the robust mean vectors estimates by group (each row refers to a group)

CovConfCases:

List of the considered configurations

grouping:

Inherited from class IdtMxE. Factor indicating the group to which each observation belongs to

ModelNames:

Inherited from class IdtE. The model acronym formed by a "N", indicating a Normal model, followed by the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; always set to "Normal" in objects of the IdtMxNDRE class

ModelConfig:

Inherited from class IdtE. Configuration case of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to FALSE in objects of class IdtMxNDRE

Ngrps:

Inherited from class IdtMxE. Number of mixture components

rawSet

A vector with the trimmed subset elements used to compute the raw (not reweighted) MCD covariance estimate for the chosen configuration.

RewghtdSet

A vector with the final trimmed subset elements used to compute the fasttle estimates.

RobMD2

A vector with the robust squared Mahalanobis distances used to select the trimmed subset.

cnp2

A vector of length two containing the consistency correction factor and the finite sample correction factor of the final estimate of the covariance matrix.

raw.cov

A matrix with the raw MCD estimator used to compute the robust squared Mahalanobis distances of RobMD2.

raw.cnp2

A vector of length two containing the consistency correction factor and the finite sample correction factor of the raw estimate of the covariance matrix.

PerfSt

A a list with the following components:
RepSteps: A list with one component by Covariance Configuration, containing a vector with the number of refinement steps performed by the fasttle algorithm by replication.
RepLogLik: A list with one component by Covariance Configuration, containing a vector with the best log-likelihood found be fasttle algorithm by replication.
StpLogLik: A list with one component by Covariance Configuration, containing a matrix with the evolution of the log-likelihoods found be fasttle algorithm by replication and refinement step.

Extends

Class IdtMxE, directly. Class IdtE, by class IdtMxE, distance 2.

Methods

No methods defined with class IdtMxNDRE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

See Also

IdtE, IdtMxE, IdtMxNDE, IdtSngNDRE, RobMxtDEst, IData

Description

IdtMxSNDE contains the results of a mixture model estimation for the Skew-Normal model, with the four different possible variance-covariance configurations.

Slots

Hmcdt:

Indicates whether we consider an homoscedastic location model (TRUE) or a general model (FALSE)

CovConfCases:

List of the considered configurations

grouping:

Inherited from class IdtMxE. Factor indicating the group to which each observation belongs to

ModelNames:

Inherited from class IdtE. The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; currently, Gaussian or Skew-Normal distributions are implemented

ModelConfig:

Inherited from class IdtE. Configuration case of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to FALSE in objects of class IdtMxSNDE

Ngrps:

Inherited from class IdtMxE. Number of mixture components

Extends

Class IdtMxE, directly. Class IdtE, by class IdtMxE, distance 2.

Methods

No methods defined with class IdtMxSNDE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IdtE, IdtMxE, IdtSngSNDE, MANOVA, IData

Description

IdtMxtNDE is an union of classes IdtMxNDE and IdtMxNDRE, containing the results of mixture Normal model parameter estimation by maximum likelihood (IdtMxNDE) or robust (IdtMxNDRE) methods.

See Also

IdtE, IdtMxE, IdtMxNDE, IdtMxNDRE

Description

IdtNandSNDE is a union of classes IdtSngNandSNDE and IdtMxNandSNDE, used for storing the estimation results of Normal and Skew-Normal modelisations for Interval Data.

Methods

coef

signature(coef = "IdtNandSNDE"): extracts parameter estimates from objects of class IdtNandSNDE

stdEr

signature(x = "IdtNandSNDE"): extracts standard errors from objects of class IdtNandSNDE

vcov

signature(x = "IdtNandSNDE"): extracts an estimate of the variance-covariance matrix of the parameters estimators for objects of class IdtNandSNDE

mean

signature(x = "IdtNandSNDE"): extracts the mean vector estimate from objects of class IdtNandSNDE

var

signature(x = "IdtNandSNDE"): extracts the variance-covariance matrix estimate from objects of class IdtNandSNDE

cor

signature(x = "IdtNandSNDE"): extracts the correlation matrix estimate from objects of class IdtNandSNDE

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IData, mle, fasttle, fulltle, MANOVA, RobMxtDEst, IdtSngNandSNDE, IdtMxNandSNDE

Description

IdtNDE is a a union of classes IdtSngNDE, IdtSngNDRE, IdtMxNDE and IdtMxNDRE, used for storing the estimation results of Normal modelizations for Interval Data.

Methods

coef

signature(coef = "IdtNDE"): extracts parameter estimates from objects of class IdtNDE

stdEr

signature(x = "IdtNDE"): extracts standard errors from objects of class IdtNDE

vcov

signature(x = "IdtNDE"): extracts an estimate of the variance-covariance matrix of the parameters estimators for objects of class IdtNDE

mean

signature(x = "IdtNDE"): extracts the mean vector estimate from objects of class IdtNDE

var

signature(x = "IdtNDE"): extracts the variance-covariance matrix estimate from objects of class IdtNDE

cor

signature(x = "IdtNDE"): extracts the correlation matrix estimate from objects of class IdtNDE

sd

signature(Idt = "IdtNDE"): extracts the standard deviation estimates from objects of class IdtNDE.

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IdtSngNDE, IdtSngNDRE, IdtMxNDE, IdtMxNDRE, IdtSNDE, IData, mle, fasttle, fulltle, MANOVA, RobMxtDEst

Description

A description of interval-valued variable outliers found by the MAINT.Data function getIdtOutl.

Slots

outliers:

A vector of indices of the interval data units flaged as outliers.

MD2:

A vector of squared robust Mahalanobis distances for all interval data units.

eta

Nominal size of the null hypothesis that a given observation is not an outlier.

RefDist

The assumed reference distributions used to find cutoffs defining the observations assumed as outliers. Alternatives are “ChiSq” and “CerioliBetaF” respectivelly for the usual Chi-squared, and the Beta and F distributions proposed by Cerioli (2010).

multiCmpCor

Whether a multicomparison correction of the nominal size (eta) for the outliers tests was performed. Alternatives are: ‘never’ – ignoring the multicomparisons and testing all entities at the ‘eta’ nominal level. ‘always’ – testing all n entitites at 1.- (1.-‘eta’^(1/n)).

NObs

Number of original observations in the original data set.

p

Number of total numerical variables (MidPoints and/or LogRanges) that may be responsible for the outliers.

h

Size of the subsets over which the trimmed likelihood was maximized when computing the robust Mahalanobis distances.

)

boolRewind

A logical vector indicanting which of the data units belong to the final trimmed subsetused to compute the tle estimates.

)

Methods

show

signature(object = "IdtOutl"): show S4 method for the IdtOutl-class.

plot

signature(x = "IdtOutl"): plot S4 methods for the IdtOutl-class.

getMahaD2

signature(x = "IdtOutl"): retrieves the vector of squared robust Mahalanobis distances for all data units.

geteta

signature(x = "IdtOutl"): retrieves the nominal size of the null hypothesis used to flag observations as outliers.

getRefDist

signature(x = "IdtOutl"): retrieves the assumed reference distributions used to find cutoffs defining the observations assumed as outliers.

getmultiCmpCor

signature(x = "IdtOutl"): retrieves the multicomparison correction used when flaging observations as outliers.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

See Also

getIdtOutl, fasttle, fulltle

Description

Plots robust Mahalanobis distances and outlier cut-offs for an object describing potential outliers in a interval-valued data set

Usage

## S4 method for signature 'IdtOutl,missing'
plot(x, scale=c("linear","log"), RefDist=getRefDist(x), eta=geteta(x), 
  multiCmpCor=getmultiCmpCor(x), ...)

Arguments

References

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.
Journal of Computational and Graphical Statistics 14, 910–927.

See Also

getIdtOutl, fasttle, fulltle

Description

Idtqda contains the results of Quadratic Discriminant Analysis for the interval data

Slots

prior:

Prior probabilities of class membership; if unspecified, the class proportions for the training set are used; if present, the probabilities should be specified in the order of the factor levels.

means:

Matrix with the mean vectors for each group

scaling:

A three-dimensional array. For each group, g, scaling[,,g] is a matrix which transforms interval-valued observations so that within-groups covariance matrix is spherical.

ldet:

Vector of half log determinants of the dispersion matrix.

lev:

Levels of the grouping factor

CovCase:

Configuration case of the variance-covariance matrix: Case 1 through Case 4

Methods

predict

signature(object = "Idtqda"): Classifies interval-valued observations in conjunction with qda.

show

signature(object = "Idtqda"): show S4 method for the Idtqda-class

CovCase

signature(object = "Idtqda"): Returns the configuration case of the variance-covariance matrix

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

qda, MANOVA, Robqda, IData

Description

IdtSNDE is a class union of classes IdtSngSNDE and IdtMxSNDE, used for storing the estimation results of Skew-Normal modelizations for Interval Data.

Methods

coef

signature(coef = "IdtSNDE"): extracts parameter estimates from objects of class IdtSNDE

stdEr

signature(x = "IdtSNDE"): extracts standard errors from objects of class IdtSNDE

vcov

signature(x = "IdtSNDE"): extracts an asymptotic estimate of the variance-covariance matrix of the paramenters estimators for objects of class IdtSNDE

mean

signature(x = "IdtSNDE"): extracts the mean vector estimate from objects of class IdtSNDE

var

signature(x = "IdtSNDE"): extracts the variance-covariance matrix estimate from objects of class IdtSNDE

cor

signature(x = "IdtSNDE"): extracts the correlation matrix estimate from objects of class IdtSNDE

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IData, mle, MANOVA, IdtSngSNDE, IdtMxSNDE, IdtNDE

Description

IdtSNgenda contains the results of discriminant analysis for the interval data, based on a general Skew-Normal model.

Slots

prior:

Prior probabilities of class membership; if unspecified, the class proportions for the training set are used; if present, the probabilities should be specified in the order of the factor levels.

ksi:

Matrix with the direct location parameter ("ksi") estimates for each group.

eta:

Matrix with the direct scaled sekwness parameter ("eta") estimates for each group.

scaling:

For each group g, scaling[,,g] is a matrix which transforms interval-valued observations so that in each group the scale-association matrix ("Omega") is spherical.

mu:

Matrix with the centred location parameter ("mu") estimates for each group.

gamma1:

Matrix with the centred sekwness parameter ("gamma1") estimates for each group.

ldet:

Vector of half log determinants of the dispersion matrix.

lev:

Levels of the grouping factor.

CovCase:

Configuration case of the variance-covariance matrix: Case 1 through Case 4

Methods

predict

signature(object = "IdtSNgenda"): Classifies interval-valued observations in conjunction with snda.

show

signature(object = "IdtSNgenda"): show S4 method for the IdtSNgenda-class

CovCase

signature(object = "IdtSNgenda"): Returns the configuration case of the variance-covariance matrix

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

MANOVA, snda, IData

Description

IdtSngNandSNDE contains the results of a single class model estimation for the Normal and the Skew-Normal distributions, with the four different possible variance-covariance configurations.

Slots

NMod:

Estimates of the single class model for the Gaussian case

SNMod:

Estimates of the single class model for the Skew-Normal case

ModelNames:

Inherited from class IdtE. The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; currently, Gaussian or Skew-Normal distributions are implemented

ModelConfig:

Inherited from class IdtE. Configuration of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Bestmodel indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to TRUE in objects of class IdtSngNandSNDE

Extends

Class IdtSngDE, directly. Class IdtE, by class IdtSngDE, distance 2.

Methods

No methods defined with class IdtSngNandSNDE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IData, IdtMxNandSNDE, mle, fasttle, fulltle

Description

Contains the results of a single class maximum likelihood estimation for the Normal distribution, with the four different possible variance-covariance configurations.

Slots

mleNmuE:

Vector with the maximum likelihood mean vectors estimates

mleNmuEse:

Vector with the maximum likelihood means' standard errors

CovConfCases:

List of the considered configurations

ModelNames:

Inherited from class IdtE. The model acronym formed by a "N", indicating a Normal model, followed by the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; always set to "Normal" in objects of the IdtSngNDE class

ModelConfig:

Inherited from class IdtE. Configuration of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Bestmodel indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to TRUE in objects of class IdtSngNDE

Extends

Class IdtSngDE, directly. Class IdtE, by class IdtSngDE, distance 2.

Methods

No methods defined with class IdtSngNDE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

IData, mle, IdtSngNDRE, IdtSngSNDE, IdtMxNDE

Description

Contains the results of a single class robust estimation for the Normal distribution, with the four different possible variance-covariance configurations.

Slots

RobNmuE:

Matrix with the maximum likelihood mean vectors estimates

CovConfCases:

List of the considered configurations

ModelNames:

Inherited from class IdtE. The model acronym formed by a "N", indicating a Normal model, followed by the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; always set to "Normal" in objects of the IdtSngNDRE class

ModelConfig:

Inherited from class IdtE. Configuration of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Bestmodel indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to TRUE in objects of class IdtSngNDRE

rawSet

A vector with the trimmed subset elements used to compute the raw (not reweighted) MCD covariance estimate for the chosen configuration.

RewghtdSet

A vector with the final trimmed subset elements used to compute the tle estimates.

RobMD2

A vector with the robust squared Mahalanobis distances used to select the trimmed subset.

cnp2

A vector of length two containing the consistency correction factor and the finite sample correction factor of the final estimate of the covariance matrix.

raw.cov

A matrix with the raw MCD estimator used to compute the robust squared Mahalanobis distances of RobMD2.

raw.cnp2

A vector of length two containing the consistency correction factor and the finite sample correction factor of the raw estimate of the covariance matrix.

PerfSt

A a list with the following components:
RepSteps: A list with one component by Covariance Configuration, containing a vector with the number of refinement steps performed by the fasttle algorithm by replication.
RepLogLik: A list with one component by Covariance Configuration, containing a vector with the best log-likelihood found be fasttle algorithm by replication.
StpLogLik: A list with one component by Covariance Configuration, containing a matrix with the evolution of the log-likelihoods found be fasttle algorithm by replication and refinement step.

Extends

Class IdtSngDE, directly. Class IdtE, by class IdtSngDE, distance 2.

Methods

No methods defined with class IdtSngNDRE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

See Also

IData, fasttle, fulltle, IdtSngNDE, IdtMxNDRE

Description

Contains the results of a single class maximum likelihood estimation for the Skew-Normal distribution, with the four different possible variance-covariance configurations.

Slots

CovConfCases:

List of the considered configurations

ModelNames:

The model acronym, indicating the model type (currently, N for Normal and SN for Skew-Normal), and the configuration Case (C1 to C4) for the covariance matrix

ModelNames:

Inherited from class IdtE. The model acronym formed by a "SN", indicating a skew-Normal model, followed by the configuration (Case 1 through Case 4)

ModelType:

Inherited from class IdtE. Indicates the model; always set to "SkewNormal" in objects of the IdtSngSNDE class

ModelConfig:

Inherited from class IdtE. Configuration case of the variance-covariance matrix: Case 1 through Case 4

NIVar:

Inherited from class IdtE. Number of interval variables

SelCrit:

Inherited from class IdtE. The model selection criterion; currently, AIC and BIC are implemented

logLiks:

Inherited from class IdtE. The logarithms of the likelihood function for the different cases

AICs:

Inherited from class IdtE. Value of the AIC criterion

BICs:

Inherited from class IdtE. Value of the BIC criterion

BestModel:

Inherited from class IdtE. Indicates the best model according to the chosen selection criterion

SngD:

Inherited from class IdtE. Boolean flag indicating whether a single or a mixture of distribution were estimated. Always set to TRUE in objects of class IdtSngSNDE

Extends

Class IdtSngDE, directly. Class IdtE, by class IdtSngDE, distance 2.

Methods

No methods defined with class IdtSngSNDE in the signature.

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

See Also

mle, IData, IdtSngNDE, IdtMxSNDE

Description

IdtSNlocda contains the results of Discriminant Analysis for the interval data, based on a location Skew-Normal model.

Slots

prior:

Prior probabilities of class membership; if unspecified, the class proportions for the training set are used; if present, the probabilities should be specified in the order of the factor levels.

ksi:

Matrix with the direct location parameter ("ksi") estimates for each group.

eta:

Vector with the direct scaled skewness parameter ("eta") estimates.

scaling:

Matrix which transforms observations to discriminant functions, normalized so that the within groups scale-association matrix ("Omega") is spherical.

mu:

Matrix with the centred location parameter ("mu") estimates for each group.

gamma1:

Vector with the centred skewness parameter ("gamma1") estimates.

N:

Number of observations.

CovCase:

Configuration case of the variance-covariance matrix: Case 1 through Case 4

Methods

predict

signature(object = "IdtSNlocda"): Classifies interval-valued observations in conjunction with snda.

show

signature(object = "IdtSNlocda"): show S4 method for the IDdtlda-class

CovCase

signature(object = "IdtSNlocda"): Returns the configuration case of the variance-covariance matrix

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

snda, MANOVA, IData

Description

lda performs linear discriminant analysis of Interval Data based on classic estimates of a mixture of Gaussian models.

Usage

## S4 method for signature 'IData'
lda(x, grouping, prior="proportions", CVtol=1.0e-5, egvtol=1.0e-10, 
  subset=1:nrow(x), CovCase=1:4, SelCrit=c("BIC","AIC"), silent=FALSE, k2max=1e6, ... )

## S4 method for signature 'IdtMxtNDE'
lda(x, prior="proportions", selmodel=BestModel(x), egvtol=1.0e-10,
  silent=FALSE, k2max=1e6, ... )

## S4 method for signature 'IdtClMANOVA'
lda( x, prior="proportions", selmodel=BestModel(H1res(x)),
  egvtol=1.0e-10, silent=FALSE, k2max=1e6, ... )

## S4 method for signature 'IdtLocNSNMANOVA'
lda( x, prior="proportions", 
  selmodel=BestModel(H1res(x)@NMod), egvtol=1.0e-10, silent=FALSE, k2max=1e6, ... )

Arguments

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

qda, snda, Roblda, Robqda, IData, IdtMxtNDE, IdtClMANOVA, IdtLocNSNMANOVA, qda, ConfMat

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

#Linear Discriminant Analysis

ChinaT.lda <- lda(ChinaT,ChinaTemp$GeoReg)
cat("Temperatures of China -- linear discriminant analysis results:\n")
print(ChinaT.lda)
ldapred <- predict(ChinaT.lda,ChinaT)$class
cat("lda Prediction results:\n")
print(ldapred )
cat("Resubstition confusion matrix:\n")
ConfMat(ChinaTemp$GeoReg,ldapred)

## Not run: 
#Estimate error rates by ten-fold cross-validation replicated 20 times  

CVlda <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=lda,CovCase=CovCase(ChinaT.lda))
summary(CVlda[,,"Clerr"])

## End(Not run)

Description

This data set consist of the lower and upper bounds of the intervals for four interval characteristics of the loans aggregated by their purpose. The original microdata is available at the Kaggle Data Science platform and consists of 887 383 loan records characterized by 75 descriptors. Among the large set of variables available, we focus on borrowers' income and account and loan information aggregated by the 14 loan purposes, wich are considered as the units of interest.

Usage

data(LoansbyPurpose_minmaxDt)

Format

A data frame containing 14 observations on the following 8 variables.

ln-inc_min

The minimum, for the current loan purpose, of natural logarithm of the self-reported annual income provided by the borrower during registration.

ln-inc_max

The maximum, for the current loan purpose, of natural logarithm of the self-reported annual income provided by the borrower during registration.

ln-revolbal_min

The minimum, for the current loan purpose, of natural logarithm of the total credit revolving balance.

ln-revolbal_max

The maximum, for the current loan purpose, of natural logarithm of the total credit revolving balance.

open-acc_min

The minimum, for the current loan purpose, of the number of open credit lines in the borrower's credit file.

open-acc_max

The maximum, for the current loan purpose, of the number of open credit lines in the borrower's credit file.

total-acc_min

The minimum, for the current loan purpose, of the total number of credit lines currently in the borrower's credit file.

total-acc_max

The maximum, for the current loan purpose, of the total number of credit lines currently in the borrower's credit file.

Source

https:www.kaggle.com/wendykan/lending-club-loan-data

Description

This data set consist of the lower and upper bounds of the intervals for four interval characteristics for 35 risk levels (from A1 to G5) of loans. The original microdata is available at the Kaggle Data Science platform and consists of 887 383 loan records characterized by 75 descriptors. Among the large set of variables available, we focus on borrowers' income and account and loan information aggregated by the 35 risk levels wich are considered as the units of interest.

Usage

data(LoansbyRiskLvs_minmaxDt)

Format

A data frame containing 35 observations on the following 8 variables.

ln-inc_min

The minimum, for the current risk category, of natural logarithm of the self-reported annual income provided by the borrower during registration.

ln-inc_max

The maximum, for the current risk category, of natural logarithm of the self-reported annual income provided by the borrower during registration.

int-rate_min

The minimum, for the current risk category, of the interest rate on the loan.

int-rate_max

The maximum, for the current risk category, of the interest rate on the loan.

open-acc_min

The minimum, for the current risk category, of the number of open credit lines in the borrower's credit file.

open-acc_max

The maximum, for the current risk category, of the number of open credit lines in the borrower's credit file.

total-acc_min

The minimum, for the current risk category, of the total number of credit lines currently in the borrower's credit file.

total-acc_max

The maximum, for the current risk category, of the total number of credit lines currently in the borrower's credit file.

Source

https:www.kaggle.com/wendykan/lending-club-loan-data

Description

This data set consist of the ten and ninety per cent quantiles of the intervals for four interval characteristics for 35 risk levels (from A1 to G5) of loans. The original microdata is available at the Kaggle Data Science platform and consists of 887 383 loan records characterized by 75 descriptors. Among the large set of variables available, we focus on borrowers' income and account and loan information aggregated by the 35 risk levels wich are considered as the units of interest.

Usage

data(LoansbyRiskLvs_qntlDt)

Format

A data frame containing 35 observations on the following 8 variables.

ln-inc_q0.10

The ten percent quantile, for the current risk category, of natural logarithm of the self-reported annual income provided by the borrower during registration.

ln-inc_q0.90

The ninety percent quantile, for the current risk category, of natural logarithm of the self-reported annual income provided by the borrower during registration.

int-rate_q0.10

The ten percent quantile, for the current risk category, of the interest rate on the loan.

int-rate_q0.90

The ninety percent quantile, for the current risk category, of the interest rate on the loan.

open-acc_q0.10

The ten percent quantile, for the current risk category, of the number of open credit lines in the borrower's credit file.

open-acc_q0.90

The ninety percent quantile, for the current risk category, of the number of open credit lines in the borrower's credit file.

total-acc_q0.10

The ten percent quantile, for the current risk category, of the total number of credit lines currently in the borrower's credit file.

total-acc_q0.90

The ninety percent quantile, for the current risk category, of the total number of credit lines currently in the borrower's credit file.

Source

https:www.kaggle.com/wendykan/lending-club-loan-data

Description

LRTest contains the results of likelihood ratio tests

Slots

ChiSq:

Value of the Chi-Square statistics corresponding to the performed test

df:

Degrees of freedom of the Chi-Square statistics

pvalue:

p-value of the Chi-Square statistics value, obtained from the Chi-Square distribution with df degrees of freedom

H0logLik:

Logarithm of the Likelihood function under the null hypothesis

H1logLik:

Logarithm of the Likelihood function under the alternative hypothesis

Methods

show

signature(object = "LRTest"): show S4 method for the LRTest-class

Author(s)

Pedro Duarte Silva <[email protected]>
Paula Brito <mpbrito.fep.up.pt>

See Also

mle, IData, ConfTests, MANOVA

Description

Function MANOVA performs MANOVA tests based on likelihood ratios allowing for both Gaussian and Skew-Normal distributions and homoscedastic or heteroscedastic setups. Methods H0res and H1res retrieve the model estimates under the null and alternative hypothesis, and method show displays the MANOVA results.

Usage

MANOVA(Sdt, grouping, Model=c("Normal","SKNormal","NrmandSKN"), CovCase=1:4, 
		SelCrit=c("BIC","AIC"), Mxt=c("Hom","Het","Loc","Gen"), 
                CVtol=1.0e-5, k2max=1e6,
		OptCntrl=list(), onerror=c("stop","warning","silentNull"), ...)

## S4 method for signature 'IdtMANOVA'
H0res(object)
## S4 method for signature 'IdtMANOVA'
H1res(object)
## S4 method for signature 'IdtMANOVA'
show(object)

Arguments

Value

An object of class IdtMANOVA, containing the estimation and test results.

See Also

IdtMANOVA, RepLOptim

Examples

#Create an Interval-Data object containing the intervals of temperatures by quarter 
# for 899 Chinese meteorological stations.
ChinaT <- IData(ChinaTemp[1:8])

#Classical (homoscedastic) MANOVA tests

ManvChina <- MANOVA(ChinaT,ChinaTemp$GeoReg)
cat("China, MANOVA by geografical regions results =\n")
print(ManvChina)

#Heteroscedastic MANOVA tests

HetManvChina <- MANOVA(ChinaT,ChinaTemp$GeoReg,Mxt="Het")
cat("China, heterocedastic MANOVA by geografical regions results =\n")
print(HetManvChina)

#Skew-Normal based MANOVA assuming the the groups differ only according to location parameters
## Not run: 

SKNLocManvChina <- MANOVA(ChinaT,ChinaTemp$GeoReg,Model="SKNormal",Mxt="Loc")
cat("China, Skew-Normal MANOVA (location model) by geografical regions results =\n")
print(SKNLocManvChina)

#Skew-Normal based MANOVA assuming the the groups may differ in all parameters

SKNGenManvChina <- MANOVA(ChinaT,ChinaTemp$GeoReg,Model="SKNormal",Mxt="Gen")
cat("China, Skew-Normal MANOVA (general model)  by geografical regions results =\n")
print(SKNGenManvChina)


## End(Not run)

Description

Function MANOVAPermTest performs a MANOVA permutation test allowing for both Gaussian and Skew-Normal distributions and homoscedastic or heteroscedastic setups.

Usage

MANOVAPermTest(MANOVAres, Sdt, grouping, nrep=200,
    Model=c("Normal","SKNormal","NrmandSKN"), CovCase=1:4,
    SelCrit=c("BIC","AIC"), Mxt=c("Hom","Het","Loc","Gen"), CVtol=1.0e-5, k2max=1e6,
    OptCntrl=list(), onerror=c("stop","warning","silentNull"), ...)

Arguments

Details

Function MANOVAPermTest performs a MANOVA permutation test allowing for both Gaussian and Skew-Normal distributions and homoscedastic or heteroscedastic setups. This test is implemented by simulating the null distribution of the MANOVA likelihood ratio statistic, using many random permutations of the observation group labels. It is intended as an alternative of the classical Chi-squares based MANOVA likelihood ratio tests, when small sample sizes cast doubt on the applicability of the Chi-squared distribution. We note that this test may be computationally intensive, in particular when used for the Skw-Normal model.

Value

the p-value of the MANOVA permutation test.

See Also

MANOVA, IdtMANOVA

Examples

## Not run: 

#Perform a MANOVA of the AbaloneIdt data set, comparing the Abalone variable means 
# according to their age 

# Create an Interval-Data object containing the Length, Diameter, Height, Whole weight, 
# Shucked weight, Viscera weight (VW), and Shell weight (SeW) of 4177 Abalones, 
# aggregated by sex and age.
# Note: The original micro-data (imported UCI Machine Learning Repository Abalone dataset) 
# is given in the AbaDF data frame, and the corresponding values of the sex by age combinations 
# is represented by the AbUnits factor. 

AbaloneIdt <- AgrMcDt(AbaDF,AbUnits)

# Create a factor with three levels (Young, Adult and Old) for Abalones with respectively 
# less than 10 rings, between 11 and 18 rings, and more than 18 rings. 

Agestrg <- substring(rownames(AbaloneIdt),first=3)
AbalClass <- factor(ifelse(Agestrg=="1-3"|Agestrg=="4-6"| Agestrg=="7-9","Young",
  ifelse(Agestrg=="10-12"|Agestrg=="13-15"| Agestrg=="16-18","Adult","Old") ) )

#Perform a classical MANOVA, computing the p-value from the asymptotic Chi-squared distribution 
# of the Wilk's lambda statistic

MANOVAres <- MANOVA(AbaloneIdt,AbalClass)
summary(MANOVAres)

#Find a finite sample p-value of the test statistic, using a permutation test.

MANOVAPermTest(MANOVAres,AbaloneIdt,AbalClass)


## End(Not run)

Description

S4 methods for function mean. These methods extract estimates of mean vectors for the models fitted to Interval Data.

Usage

## S4 method for signature 'IdtNDE'
mean(x)
## S4 method for signature 'IdtSNDE'
mean(x)
## S4 method for signature 'IdtNandSNDE'
mean(x)
## S4 method for signature 'IdtMxNDE'
mean(x)
## S4 method for signature 'IdtMxSNDE'
mean(x)

Arguments

Value

For the IdtNDE, IdtSNDE and IdtNandSNDE methods or IdtMxNDE, IdtMxSNDE methods with slot “Hmcdt” equal to TRUE: a matrix with the estimated correlations.
For the IdtMxNDE, and IdtMxSNDE methods with slot “Hmcdt” equal to FALSE: a three-dimensional array with a matrix with the estimated correlations for each group at each level of the third dimension.

See Also

sd var cor

Description

Performs maximum likelihood estimation for parametric models of interval data

Usage

## S4 method for signature 'IData'
mle(Sdt, Model="Normal", CovCase="AllC", SelCrit=c("BIC","AIC"), 
  k2max=1e6, OptCntrl=list(), ...)

Arguments

References

Azzalini, A. and Dalla Valle, A. (1996), The multivariate skew-normal distribution. Biometrika 83(4), 715–726.

Brito, P., Duarte Silva, A. P. (2012): "Modelling Interval Data with Normal and Skew-Normal Distributions". Journal of Applied Statistics, Volume 39, Issue 1, 3-20.

See Also

IData, RepLOptim

Examples

# Create an Interval-Data object containing the intervals of temperatures by quarter 
# for 899 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8])

# Estimate parameters by maximum likelihood assuming a Gaussian distribution

ChinaE <- mle(ChinaT)
cat("China maximum likelhiood estimation results =\n")
print(ChinaE)
cat("Standard Errors of Estimators:\n")
print(stdEr(ChinaE))

Description

A interval-valued data set containing 142 units and four interval-valued variables (dep_delay, arr_delay, air_time and distance), created from from the flights data set in the R package nycflights13 (on-time data for all flights that departed the JFK, LGA or EWR airports in 2013), after removing all rows with missing observations, and aggregating by month and carrier.

Usage

data(nycflights)

Format

FlightsDF: A data frame containing the original 327346 valid (i.e. with non missing values) flights from the nycflights13 package, described by the 4 variables: dep_delay, arr_delay, air_time and distance.
FlightsUnits: A factor with 327346 observations and 142 levels, indicating the month by carrier combination to which each orginal flight belongs to.
FlightsIdt: An IData object with 142 observations and 4 interval-valued variables, describing the intervals formed by agregating the FlightsDF microdata by the 0.05 and 0.95 quantiles of the subsamples formed by FlightsUnits factor.

Description

Method pcoordplot displays a parallel coordinates plot, representing the results stored in an IdtMclust-method object.

Usage

## S4 method for signature 'IdtMclust'
pcoordplot(x,title="Parallel Coordinate Plot",
Seq=c("AllMidP_AllLogR","MidPLogR_VarbyVar"), model ="BestModel", legendpar=list(), ...)

Arguments

See Also

IdtMclust, Idtmclust, plotInfCrt

Examples

## Not run: 

# Create an Interval-Data object containing the intervals of loan data
# (from the Kaggle Data Science platform) aggregated by loan purpose

LbyPIdt <- IData(LoansbyPurpose_minmaxDt,
                 VarNames=c("ln-inc","ln-revolbal","open-acc","total-acc")) 


#Fit homoscedastic Gaussian mixtures with up to ten components

mclustres <- Idtmclust(LbyPIdt,G=1:10)
plotInfCrt(mclustres,legpos="bottomright")

#Display the results of the best mixture according to the BIC

pcoordplot(mclustres)
pcoordplot(mclustres,model="HomG6C1")
pcoordplot(mclustres,model="HomG4C1")



## End(Not run)

Description

S4 methods for function plot. As in the generic plot S3 ‘graphics’ method, these methods plot Interval-valued data contained in IData objects.

Usage

## S4 method for signature 'IData,IData'
plot(x, y, type=c("crosses","rectangles"), append=FALSE, ...)
## S4 method for signature 'IData,missing'
plot(x, casen=NULL, layout=c("vertical","horizontal"), append=FALSE, ...)

Arguments

See Also

IData

Examples

## Not run: 

# Create an Interval-Data object containing the Length, Diameter, Height, Whole weight, 
# Shucked weight, Viscera weight (VW), and Shell weight (SeW) of 4177 Abalones, 
# aggregated by sex and age.
# Note: The original micro-data (imported UCI Machine Learning Repository Abalone dataset) 
# is given in the AbaDF data frame, and the corresponding values of the sex by age combinations 
# is represented by the AbUnits factor. 

AbaloneIdt <- AgrMcDt(AbaDF,AbUnits)

# Dispaly a plot of the Length versus the Whole_weight interval variables

plot(AbaloneIdt[,"Length"],AbaloneIdt[,"Whole_weight"])
plot(AbaloneIdt[,"Length"],AbaloneIdt[,"Whole_weight"],type="rectangles")

# Display the Abalone lengths using different colors to distinguish the Abalones age 
# (measured by the number of rings)   

# Create a factor with three levels (Young, Adult and Old) for Abalones with 
# respectively less than 10 rings, between 11 and 18 rings, and more than 18 rings. 

Agestrg <- substring(rownames(AbaloneIdt),first=3)
AbalClass <- factor(ifelse(Agestrg=="1-3"|Agestrg=="4-6"| Agestrg=="7-9","Young",
  ifelse(Agestrg=="10-12"|Agestrg=="13-15"| Agestrg=="16-18","Adult","Old") ) )

plot(AbaloneIdt[AbalClass=="Young","Length"],col="blue",layout="horizontal") 
plot(AbaloneIdt[AbalClass=="Adult","Length"],col="green",layout="horizontal",append=TRUE) 
plot(AbaloneIdt[AbalClass=="Old","Length"],col="red",layout="horizontal",append=TRUE) 
legend("bottomleft",legend=c("Young","Adult","Old"),col=c("blue","green","red"),lty=1)



## End(Not run)

Description

Method plotInfCrt displays a plot representing the values of an appropriate information criterion (currently either BIC or AIC) for the models whose results are stored in an IdtMclust-method object. A supplementary short output message prints the values of the chosen criterion for the 'nprin' best models.

Usage

## S4 method for signature 'IdtMclust'
plotInfCrt(object, crt=object@SelCrit, legpos="right", nprnt=5,
  legendout=TRUE, outlegsize="adjstoscreen", outlegdisp="adjstoscreen", 
  legendpar=list(), ...)

Arguments

See Also

IdtMclust, Idtmclust, pcoordplot

Examples

## Not run: 

# Create an Interval-Data object containing the intervals of loan data
# (from the Kaggle Data Science platform) aggregated by loan purpose

LbyPIdt <- IData(LoansbyPurpose_minmaxDt,
                 VarNames=c("ln-inc","ln-revolbal","open-acc","total-acc")) 

#Fit homoscedastic and heteroscedastic mixtures up to Gaussian mixtures with up to seven components

mclustres <- Idtmclust(LbyPIdt,G=1:7,Mxt="HomandHet")

#Compare de model fit according to the BIC

plotInfCrt(mclustres,legpos="bottomleft")

#Display the results of the best three mixtures according to the BIC

summary(mclustres,parameters=TRUE,classification=TRUE)
pcoordplot(mclustres)
summary(mclustres,parameters=TRUE,classification=TRUE,model="HetG2C2")
summary(mclustres,parameters=TRUE,classification=TRUE,model="HomG6C1")
pcoordplot(mclustres,model="HomG6C1")



## End(Not run)

Description

qda performs quadratic discriminant analysis of Interval Data based on classic estimates of a mixture of Gaussian models.

Usage

## S4 method for signature 'IData'
qda( x, grouping, prior="proportions", CVtol=1.0e-5, subset=1:nrow(x),
  CovCase=1:4, SelCrit=c("BIC","AIC"), silent=FALSE, k2max=1e6, ... )

## S4 method for signature 'IdtMxtNDE'
qda(x, prior="proportions", selmodel=BestModel(x), silent=FALSE, 
  k2max=1e6, ... )

## S4 method for signature 'IdtHetNMANOVA'
qda( x, prior="proportions", selmodel=BestModel(H1res(x)), 
  silent=FALSE, k2max=1e6, ... )

## S4 method for signature 'IdtGenNSNMANOVA'
qda( x, prior="proportions", 
  selmodel=BestModel(H1res(x)@NMod), silent=FALSE, k2max=1e6, ... )

Arguments

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

See Also

lda, snda, Roblda, Robqda, IData, IdtMxtNDE, IdtHetNMANOVA, IdtGenNSNMANOVA, ConfMat

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

#Quadratic Discriminant Analysis

ChinaT.qda <- qda(ChinaT,ChinaTemp$GeoReg)
cat("Temperatures of China -- qda discriminant analysis results:\n")
print(ChinaT.qda)
cat("Resubstition confusion matrix:\n")
ConfMat(ChinaTemp$GeoReg,predict(ChinaT.qda,ChinaT)$class)

## Not run: 
#Estimate error rates by ten-fold cross-validation replicated 20 times  

CVqda <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=qda,CovCase=CovCase(ChinaT.qda))
summary(CVqda[,,"Clerr"])


## End(Not run)

Description

p-quantiles of the Hardin and Rocke (2005) scaled F distribution for squared Mahalanobis distances based on raw MCD covariance estimators

Usage

qHardRoqF(p, nobs, nvar, h=floor((nobs+nvar+1)/2), adj=TRUE, 
  lower.tail=TRUE, log.p=FALSE)

Arguments

Value

The quantile of the appropriate scaled F distribution.

References

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

See Also

fasttle, fulltle

Description

‘RepLOptim’ Tries to minimize a function calling local optimizers several times from different random starting points.

Usage

RepLOptim(start, parsd, fr, gr=NULL, inphess=NULL, ..., method="nlminb",
	 lower=NULL, upper=NULL, rethess=FALSE, parmstder=FALSE, control=list())

Arguments

Details

‘RepLOptim’ Tries to minimize a function by calling local optimizers several times from different starting points. The starting point used in the first call the the local optimizer is the value of the argument ‘start’. Subsquent calls use starting points generated from uniform distributions of independent variates with means equal to the current best parameter values and standard deviations equal to the values of the argument ‘parsd’. If parameter bounds are specified and the uniform limits implied by ‘parsd’ violate those bounds, these limits are replaced by the corresponding bounds.

The choice of the local optimizer is made by value of the ‘method’ argument. This argument can be a function object implementing the optimizer or a string describing an available R method. In the latter case current alternatives are: “nlminb” (default) for the ‘nlminb’ port routine, “nlm” for the ‘nlm’ function and “Nelder-Mead”, “L-BFGS-B”, “CG”, “L-BFGS-B” and “SANN” for the corresponding methods of the ‘optim’ function.

Arguments for controling the behaviour of the local optimizer can be specified as components of control list. This list can include any of the following components:

maxrepet

Maximum time of repetions of the same minimum objective value, before RepLOptim is stoped and the current best solution is returned. By default set to 2.

maxnoimprov

Maximum number of times the local optimizer is called without improvements in the minimum objective value, before RepLOptim is stopped and the current best solution is returned. By default set to 50.

maxreplic

Maximum number of times the local optimizer is called and returns a valid solution before RepLOptim is stoped and the current best solution is returned. By default set to 250.

allrep

Total maximum number of replications (including those leading to non-valid solutions) performed. By default equals ten times the value of maxreplic. Ignored when objbnd is set to ‘Inf’.

maxiter

Maximum number of iterations performed in each call to the local optimizer. By default set to 500 except with the “SANN” mehtod, when by default is set to 1500.

maxeval

Maximum number of function evaluations (nlminb method only) performed in each call to the nlminb optimizer. By defaults set to 1000.

RLOtol

The relative convergence tolerance of the local optimizer. The local optimizer stops if it is unable to reduce the value by a factor of ‘RLOtol *(abs(val) + reltol)’ at a step. Ignored when method is set to “nlm”. By default set to the square root of the computer precision, i.e. to ‘sqrt(.Machine$double.eps)’.

HesEgtol

Numerical tolerance used to ensure that the hessian is non-singular. If the last eigenvalue of the hessian is positive but the ratio between it and the first eigenvalue is below HesEgtol the hessian is considered to be semi-definite and the parameter assymptotic standard errors are not computed. By default set to the square root of the computer precision, i.e. to ‘sqrt(.Machine$double.eps)’.

objbnd

Upper bound for the objective. Only solutions leading to objective values below objbnd are considered as valid.

Value

A list with the following components:

Author(s)

A. Pedro Duarte Silva

Description

Roblda and Robqda perform linear and quadratic discriminant analysis of Interval Data based on robust estimates of location and scatter.

Usage

## S4 method for signature 'IData'
Roblda( x, grouping, prior="proportions", CVtol=1.0e-5, egvtol=1.0e-10,
  subset=1:nrow(x), CovCase=1:4, SelCrit=c("BIC","AIC"), silent=FALSE, 
  CovEstMet=c("Pooled","Globdev"), SngDMet=c("fasttle","fulltle"), k2max=1e6,
  Robcontrol=RobEstControl(), ... )

## S4 method for signature 'IData'
Robqda( x, grouping, prior="proportions", CVtol=1.0e-5, 
  subset=1:nrow(x), CovCase=1:4, SelCrit=c("BIC","AIC"), silent=FALSE,
  SngDMet=c("fasttle","fulltle"), k2max=1e6, Robcontrol=RobEstControl(), ... )

Arguments

References

Duarte Silva, A.P. and Brito, P. (2015), Discriminant analysis of interval data: An assessment of parametric and distance-based approaches. Journal of Classification 39(3), 516–541.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

See Also

lda, qda, snda, IData, RobEstControl,codeConfMat

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

#Robust Linear Discriminant Analysis

## Not run: 

ChinaT.rlda <- Roblda(ChinaT,ChinaTemp$GeoReg)
cat("Temperatures of China -- robust lda discriminant analysis results:\n")
print(ChinaT.rlda)
cat("Resubstition confusion matrix:\n")
ConfMat(ChinaTemp$GeoReg,predict(ChinaT.rlda,ChinaT)$class)

#Estimate error rates by ten-fold cross-validation with 5 replications 

CVrlda <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=Roblda,CovCase=CovCase(ChinaT.rlda),
   CVrep=5)
summary(CVrlda[,,"Clerr"])

#Robust Quadratic Discriminant Analysis

ChinaT.rqda <- Robqda(ChinaT,ChinaTemp$GeoReg)
cat("Temperatures of China -- robust qda discriminant analysis results:\n")
print(ChinaT.rqda)
cat("Resubstition confusion matrix:\n")
ConfMat(ChinaTemp$GeoReg,predict(ChinaT.rqda,ChinaT)$class)

#Estimate error rates by ten-fold cross-validation with 5 replications 

CVrqda <- DACrossVal(ChinaT,ChinaTemp$GeoReg,TrainAlg=Robqda,CovCase=CovCase(ChinaT.rqda),
   CVrep=5)
summary(CVrqda[,,"Clerr"])


## End(Not run)

Description

This function will create a control object of class RobEstControl containing the control parameters for the robust estimation functions fasttle, RobMxtDEst, Roblda and Robqda.

Usage

RobEstControl(alpha=0.75, nsamp=500,  seed=NULL, trace=FALSE, use.correction=TRUE,
  ncsteps=200, getalpha="TwoStep", rawMD2Dist="ChiSq", MD2Dist="ChiSq", eta=0.025,
  multiCmpCor="never",  getkdblstar="Twopplusone", outlin="MidPandLogR", 
  trialmethod="simple", m=1, reweighted=TRUE, k2max=1e6, otpType="SetMD2andEst")

Arguments

Value

A RobEstControl object

References

Brito, P., Duarte Silva, A. P. (2012): "Modelling Interval Data with Normal and Skew-Normal Distributions". Journal of Applied Statistics, Volume 39, Issue 1, 3-20.

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

Todorov V. and Filzmoser P. (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software 32(3), 1–47.

See Also

RobEstControl, fasttle, RobMxtDEst, Roblda, Robqda

Description

This class extends the CovControlMcd class and contains control parameters for the robust estimation of parametric interval data models.

Objects from the Class

Objects can be created by calls of the form new("RobEstControl", ...) or by calling the constructor-function RobEstControl.

Slots

alpha:

Inherited from class "CovControlMcd". Numeric parameter controlling the size of the subsets over which the trimmed likelihood is maximized; roughly alpha*nrow(Sdt) observations are used for computing the trimmed likelihood. Allowed values are between 0.5 and 1. Note that when argument ‘getalpha’ is set to “TwoStep” the final value of ‘alpha’ is estimated by a two-step procedure and the value of argument ‘alpha’ is only used to specify the size of the samples used in the first step.

nsamp:

Inherited from class "CovControlMcd". Number of subsets used for initial estimates.

scalefn:

Inherited from class "CovControlMcd" and not used in the package ‘Maint.Data.’

maxcsteps:

Inherited from class "CovControlMcd" and not used in the package ‘Maint.Data.’

seed:

Inherited from class "CovControlMcd". Starting value for random generator. Default is seed = NULL.

use.correction:

Inherited from class "CovControlMcd". Whether to use finite sample correction factors. Default is use.correction=TRUE.

trace, tolSolve:

Inherited from class "CovControl".

ncsteps:

The maximum number of concentration steps used each iteration of the fasttle algorithm.

getalpha:

Argument specifying if the ‘alpha’ parameter (roughly the percentage of the sample used for computing the trimmed likelihood) should be estimated from the data, or if the value of the argument ‘alpha’ should be used instead. When set to “TwoStep”, ‘alpha’ is estimated by a two-step procedure with the value of argument ‘alpha’ specifying the size of the samples used in the first step. Otherwise, with the value of argument ‘alpha’ is used directly.

rawMD2Dist:

The assumed reference distribution of the raw MCD squared distances, which is used to find to cutoffs defining the observations kept in one-step reweighted MCD estimates. Alternatives are ‘ChiSq’,‘HardRockeAsF’ and ‘HardRockeAdjF’, respectivelly for the usual Chi-squared, and the asymptotic and adjusted scaled F distributions proposed by Hardin and Rocke (2005).

MD2Dist:

The assumed reference distributions used to find cutoffs defining the observations assumed as outliers. Alternatives are “ChiSq” and “CerioliBetaF” respectivelly for the usual Chi-squared, and the Beta and F distributions proposed by Cerioli (2010).

eta:

Nominal size of the null hypothesis that a given observation is not an outlier. Defines the raw MCD Mahalanobis distances cutoff used to choose the observations kept in the reweightening step.

multiCmpCor:

Whether a multicomparison correction of the nominal size (eta) for the outliers tests should be performed. Alternatives are: ‘never’ – ignoring the multicomparisons and testing all entities at ‘eta’. ‘always’ – testing all n entitites at 1.- (1.-‘eta’^(1/n)); and ‘iterstep’ – as suggested by Cerioli (2010), make an initial set of tests using the nominal size 1.- (1-‘eta’^(1/n)), and if no outliers were detected stop. Otherwise, make a second step testing for outliers at ‘eta’.

getkdblstar:

Argument specifying the size of the initial small (in order to minimize the probability of outliers) subsets. If set to the string “Twopplusone” (default) the initial sets have twice the number of interval-value variables plus one (i.e., they are the smaller samples that lead to a non-singular covariance estimate). Otherwise, an integer with the size of the initial sets.

k2max:

Maximal allowed l2-norm condition number for correlation matrices. Correlation matrices with condition number above k2max are considered to be numerically singular, leading to degenerate results.

outlin:

The type of outliers to be consideres. “MidPandLogR” if outliers may be present in both MidPpoints and LogRanges, “MidP” if outliers are only present in MidPpoints, or “LogR” if outliers are only present in LogRanges.

trialmethod:

The method to find a trial subset used to initialize each replication of the fasttle algorithm. The current options are “simple” (default) that simply selects ‘kdblstar’ observations at random, and “Poolm” that divides the original sample into ‘m’ non-overlaping subsets, applies the ‘simple trial’ and the refinement methods to each one of them, and merges the results into a trial subset.

m:

Number of non-overlaping subsets used by the trial method when the argument of ‘trialmethod’ is set to 'Poolm'.

reweighted:

Should a (Re)weighted estimate of the covariance matrix be used in the computation of the trimmed likelihood or just a “raw” covariance estimate; default is (Re)weighting.

otpType:

The amount of output returned by fasttle. Current options are “OnlyEst” (default) where only an ‘IdtE’ object with the fasttle estimates is returned, “SetMD2andEst” which returns a list with an ‘IdtE’ object of fasttle estimates, a vector with the final trimmed subset elements used to compute these estimates and the corresponding robust squared Mahalanobis distances, and “SetMD2EstandPrfSt” wich returns a list with the previous three components plust a list of some performance statistics concerning the algorithm execution.

Extends

Class CovControlMcd, directly. Class CovControl by CovControlMcd, distance 2.

Methods

No methods defined with class "RobEstControl" in the signature.

References

Cerioli, A. (2010), Multivariate Outlier Detection with High-Breakdown Estimators. Journal of the American Statistical Association 105 (489), 147–156.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Hardin, J. and Rocke, A. (2005), The Distribution of Robust Distances. Journal of Computational and Graphical Statistics 14, 910–927.

Todorov V. and Filzmoser P. (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software 32(3), 1–47.

See Also

RobEstControl, fasttle, RobMxtDEst, Roblda, Robqda

Description

RobMxtDEst estimates mixtures of distribution for interval-valued data using robust methods.

Usage

## S4 method for signature 'IData'
RobMxtDEst(Sdt, grouping, Mxt=c("Hom","Het"), CovEstMet=c("Pooled","Globdev"),
    CovCase=1:4, SelCrit=c("BIC","AIC"), Robcontrol=RobEstControl(),
    l1medpar=NULL, ...)

Arguments

Value

An object of class IdtMxNDRE, containing the estimation results.

References

Brito, P., Duarte Silva, A. P. (2012), Modelling Interval Data with Normal and Skew-Normal Distributions. Journal of Applied Statistics 39(1), 3–20.

Duarte Silva, A.P., Filzmoser, P. and Brito, P. (2017), Outlier detection in interval data. Advances in Data Analysis and Classification, 1–38.

Todorov V. and Filzmoser P. (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software 32(3), 1–47.

See Also

IdtMxNDRE, RobEstControl.

Examples

# Create an Interval-Data object containing the intervals for 899 observations 
# on the temperatures by quarter in 60 Chinese meteorological stations.

ChinaT <- IData(ChinaTemp[1:8],VarNames=c("T1","T2","T3","T4"))

## Not run: 

# Estimate robustly an homoscedastic mixture, with mixture components defined by regions

ChinaHomMxtRobE <- RobMxtDEst(ChinaT,ChinaTemp$GeoReg)
print(ChinaHomMxtRobE)

# Estimate robustly an heteroscedastic mixture, with mixture components defined by regions

ChinaHetMxtRobE <- RobMxtDEst(ChinaT,ChinaTemp$GeoReg,Mxt="Het")
print(ChinaHetMxtRobE)



## End(Not run)